1. Technical Field
The present disclosure relates to image analysis, and more particularly to machine learning, clustering and object segmentation within images.
2. Discussion of Related Art
The Mumford-Shah functional has had a major impact on a variety of image analysis problems including image segmentation and filtering and, despite being introduced over two decades ago, it is still in widespread use. Present day optimization of the Mumford-Shah functional is predominated by active contour methods. Unfortunately, these formulations necessitate optimization of the contour by evolving via gradient descent, which is known for its overdependence on initialization and the tendency to produce undesirable local minima.
The Mumford-Shah functional was devised to formulate the problem of finding piecewise smooth reconstructions of functions (e.g., images) as an optimization problem [1]. Optimizing the Mumford-Shah functional involves determining both a function and a contour across which smoothness is not penalized. Unfortunately, since smoothness of the reconstruction is not enforced across the contour and since the contour is variable in the optimization, the functional is not easily minimized using classical calculus of variations.
Given a fixed contour it is possible to solve for the optimal reconstruction function by solving an elliptic PDE with Neumann boundary conditions. Additionally, given a fixed piecewise smooth reconstruction function, it is possible to determine at each point on the contour, the direction and speed that the contour should move to decrease the functional as quickly as possible. Thus, most methods for solving the Mumford-Shah functional involve alternating optimization of the reconstruction function and the contour. The results of performing this style of optimization are well known and achieve satisfactory results that are usable for different imaging applications. Unfortunately, this optimization of the Mumford-Shah functional using contour evolution techniques (typically implemented with level sets) is slow primarily due to the small steps that the contour must take at each iteration. This slowness is exacerbated by the fact that a very small perturbation of the contour can have a relatively large effect on the optimal reconstruction function. Additionally, these traditional methods often require many implementation choices (e.g., implementation parameters) and the result of these choices may cause differences in the final result.
Although new functionals for segmentaton/filtering continue to be developed, the Mumford-Shah functional is still very widely used and optimized with level set methods. In addition to applications, recent work in the computer vision community has also continued to address theoretical aspects of the Mumford-Shah functional and its optimization.
Practical energy minimization problems formulated on a finite series of variables can often be solved efficiently using combinatorial (graph-based) algorithms. Furthermore, because of the well established equivalence between the standard operators of multidimensional calculus and certain combinatorial operators, it is possible to rewrite many PDEs formulated in RN equivalently on a cell complex (graph). By reformulating the conventional, continuous, PDE on a graph it becomes straightforward to apply the arsenal of combinatorial optimization techniques to efficiently solve these variational problems.
Therefore, a need exists for an improved method of determining a Mumford-Shah functional.